Discrete behavior of Seshadri constants on surfaces

Working over , we show that, apart possibly from a unique limit point, the possible values of multi-point Seshadri constants for general points on smooth projective surfaces form a discrete set. In addition to its theoretical interest, this result is of practical value, which we demonstrate by giving significantly improved explicit lower bounds for Seshadri constants on and new results about ample divisors on blow ups of at general points.     

Higher syzygies of hyperelliptic curves

Let X be a hyperelliptic curve of arithmetic genus g and let f:X→P¹ be the hyperelliptic involution map of X. In this paper we study higher syzygies of linearly normal embeddings of X of degree d≤2g. Note that the minimal free resolution of X of degree ≥2g+1 is already completely known. Let A=f^∗O_P¹(1), and let L be a very ample line bundle on X of degree d≤2g. For , we call the pair (m,d−2m)the factorization type ofL. Our main result is that the Hartshorne-Rao module and the graded Betti numbers of the linearly normal curve embedded by |L| are precisely determined by the factorization type of L.     

Essentialité dans les bases additives

In this article we study the notion of essential subset of an additive basis, that is to say the minimal finite subsets P of a basis A such that A−P does not remains a basis. The existence of an essential subset for a basis is equivalent for this basis to be included, for almost all elements, in an arithmetic non-trivial progression. We show that for every basis A there exists an arithmetic progression with a biggest common difference containing A. Having this common difference a we are able to give an upper bound to the number of essential subsets of A: this is the radical's length of a (in particular there is always many finite essential subsets in a basis). In the case of essential subsets of cardinality 1 (essential elements) we introduce a way to “dessentialize” a basis. As an application, we definitively complete the result of Deschamps and Grekos about the majoration of essential elements of a basis by showing that for all basis A of order h, the number s of essential elements of A satisfy where , and we show that this inequality is best possible.     

Borel-plus-powers monomial ideals

Let S=K[x₁,…,x_n] be a standard graded polynomial ring over a field K. In this paper, we show that the lex-plus-powers ideal has the largest graded Betti numbers among all Borel-plus-powers monomial ideals with the same Hilbert function. In addition in the case of characteristic 0, by using this result, we prove the lex-plus-powers conjecture for graded ideals containing , where p is a prime number.     

Families of low dimensional determinantal schemes

A scheme X⊂Pⁿ of codimension c is called standard determinantal if its homogeneous saturated ideal can be generated by the t×t minors of a homogeneous t×(t+c−1) matrix (f_ij). Given integers a₀≤a₁≤?≤a_t+c−2 and b₁≤?≤b_t, we denote by the stratum of standard determinantal schemes where f_ij are homogeneous polynomials of degrees a_j−b_i and is the Hilbert scheme (if n−c>0, resp. the postulation Hilbert scheme if n−c=0).Focusing mainly on zero and one dimensional determinantal schemes we determine the codimension of in and we show that is generically smooth along under certain conditions. For zero dimensional schemes (only) we find a counterexample to the conjectured value of appearing in Kleppe and Miró-Roig (2005) [25].     

Invariant hypersurfaces for derivations in positive characteristic

Let A be an integral k-algebra of finite type over an algebraically closed field k of characteristic p>0. Given a collection D of k-derivations on A, that we interpret as algebraic vector fields on , we study the group spanned by the hypersurfaces V(f) of X invariant under D modulo the rational first integrals of D. We prove that this group is always a finite dimensional F_p-vector space, and we give an estimate for its dimension. This is to be related to the results of Jouanolou and others on the number of hypersurfaces invariant under a foliation of codimension 1. As a application, given a k-algebra B between A^p and A, we show that the kernel of the pull-back morphism is a finite F_p-vector space. In particular, if A is a UFD, then the Picard group of B is finite.     

The Farahat-Higman ring of wreath products and Hilbert schemes

We study the structure constants of the class algebra of the wreath products Γ_n associated to an arbitrary finite group Γ with respect to the basis of conjugacy classes. We show that a suitable filtration on gives rise to the graded ring with non-negative integer structure constants independent of n (some of which are computed), which are then encoded in a Farahat-Higman ring . The real conjugacy classes of Γ come to play a distinguished role and are treated in detail in the case when Γ is a subgroup of . The above results provide new insight to the cohomology rings of Hilbert schemes of points on a quasi-projective surface X.     

Basic deformation theory of smooth formal schemes

We provide the main results of a deformation theory of smooth formal schemes as defined in [L. Alonso Tarrío, A. Jeremías López, M. Pérez Rodríguez, Infinitesimal lifting and Jacobi criterion for smoothness on formal schemes, Comm. Algebra 35 (2007) 1341-1367]. Smoothness is defined by the local existence of infinitesimal liftings. Our first result is the existence of an obstruction in a certain Ext¹ group whose vanishing guarantees the existence of global liftings of morphisms. Next, given a smooth morphism f₀:X₀→Y₀ of noetherian formal schemes and a closed immersion Y₀?Y given by a square zero ideal I, we prove that the set of isomorphism classes of smooth formal schemes lifting X₀ over Y is classified by and that there exists an element in which vanishes if and only if there exists a smooth formal scheme lifting X₀ over Y.     

Upper bounds for the order of an additive basis obtained by removing a finite subset of a given basis

Let A be an additive basis of order h and X be a finite nonempty subset of A such that the set A?X is still a basis. In this article, we give several upper bounds for the order of A?X in function of the order h of A and some parameters related to X and A. If the parameter in question is the cardinality of X, Nathanson and Nash already obtained some of such upper bounds, which can be seen as polynomials in h with degree (|X|+1). Here, by taking instead of the cardinality of X the parameter defined by , we show that the order of A?X is bounded above by . As a consequence, we deduce that if X is an arithmetic progression of length ?3, then the upper bounds of Nathanson and Nash are considerably improved. Further, by considering more complex parameters related to both X and A, we get upper bounds which are polynomials in h with degree only 2.     

On the automorphisms of the spectral completion of the algebraic numbers field

Let be the absolute Galois group of Q and let A=C(G,C) be the Banach algebra of all continuous functions defined on G with values in C. Let be the conjugation automorphism of C and let B be the R-Banach subalgebra of A consisting of continuous functions f such that for all σ∈G. Let ‖x‖=sup{|σ(x)|:σ∈G} be the spectral norm on and let be the spectral completion of . Using a canonical isometry between and B we study the structure of the group of R-algebras automorphisms of and the structure of its subgroup of all automorphisms of which when restricted to give rise to elements of G. We introduce a topology on and prove that this last one is homeomorphic and group isomorphic to G.     

Hilbert curves of polarized varieties

Let X be a normal Gorenstein complex projective variety. We introduce the Hilbert variety V_X associated to the Hilbert polynomial χ(x₁L₁+?+x_ρL_ρ), where L₁,…,L_ρ is a basis of , ρ being the Picard number of X, and x₁,…,x_ρ are complex variables. After studying general properties of V_X we specialize to the Hilbert curve of a polarized variety (X,L), namely the plane curve of degree dim(X) associated to χ(xK_X+yL). Special emphasis is given to the case of polarized threefolds.     

On face numbers of manifolds with symmetry

Necessary conditions on the face numbers of Cohen-Macaulay simplicial complexes admitting a proper action of the cyclic group of a prime order are given. This result is extended further to necessary conditions on the face numbers and the Betti numbers of Buchsbaum simplicial complexes with a proper -action. Adin's upper bounds on the face numbers of Cohen-Macaulay complexes with symmetry are shown to hold for all (d−1)-dimensional Buchsbaum complexes with symmetry on n?3d−2 vertices. A generalization of Kühnel's conjecture on the Euler characteristic of 2k-dimensional manifolds and Sparla's analog of this conjecture for centrally symmetric 2k-manifolds are verified for all 2k-manifolds on n?6k+3 vertices. Connections to the Upper Bound Theorem are discussed and its new version for centrally symmetric manifolds is established.     

Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers

We define and investigate extension groups in the context of Arakelov geometry. The “arithmetic extension groups” we introduce are extensions by groups of analytic types of the usual extension groups attached to OX-modules F and G over an arithmetic scheme X. In this paper, we focus on the first arithmetic extension group - the elements of which may be described in terms of admissible short exact sequences of hermitian vector bundles over X - and we especially consider the case when X is an “arithmetic curve”, namely the spectrum SpecOK of the ring of integers in some number field K. Then the study of arithmetic extensions over X is related to old and new problems concerning lattices and the geometry of numbers.Namely, for any two hermitian vector bundles and over X:=SpecOK, we attach a logarithmic size to any element α of , and we give an upper bound on in terms of slope invariants of and . We further illustrate this notion by relating the sizes of restrictions to points in P¹(Z) of the universal extension over to the geometry of PSL₂(Z) acting on Poincaré's upper half-plane, and by deducing some quantitative results in reduction theory from our previous upper bound on sizes. Finally, we investigate the behaviour of size by base change (i.e., under extension of the ground field K to a larger number field K^′): when the base field K is Q, we establish that the size, which cannot increase under base change, is actually invariant when the field K^′ is an abelian extension of K, or when is a direct sum of root lattices and of lattices of Voronoi's first kind.The appendices contain results concerning extensions in categories of sheaves on ringed spaces, and lattices of Voronoi's first kind which might also be of independent interest.     

The central simple modules of Artinian Gorenstein algebras

Let A be a standard graded Artinian K-algebra, with char K=0. We prove the following.

1.

A has the Weak Lefschetz Property (resp. Strong Lefschetz Property) if and only if has the Weak Lefschetz Property (resp. Strong Lefschetz Property) for some linear form z of A.

2.

If A is Gorenstein, then A has the Strong Lefschetz Property if and only if there exists a linear form z of A such that all central simple modules of (A,z) have the Strong Lefschetz Property.

As an application of these theorems, we give some new classes of Artinian complete intersections with the Strong Lefschetz Property.     

On the independence of Heegner points associated to distinct quadratic imaginary fields

Let E/Q be an elliptic curve with no CM and a fixed modular parametrization and let be Heegner points attached to the rings of integers of distinct quadratic imaginary fields k₁,…,kr. We prove that if the odd parts of the class numbers of k₁,…,kr are larger than a constant C=C(E,ΦE) depending only on E and ΦE, then the points P₁,…,Pr are independent in .